Solved landmarkTopology · Undergraduate → research

The Poincaré conjecture

What can loops tell us about a whole space?

Reviewed: 2026-10-20Editorial introduction · not original research

M^3\text{ closed, simply connected}\ \Longrightarrow\ M^3\cong S^3

01 / The question

In three dimensions, the conjecture characterizes the sphere using simple connectedness. Perelman’s work resolved it, and Clay lists it among the solved Millennium Prize Problems.

02 / Why it is hard

The three-sphere is a three-dimensional space, not the ordinary two-dimensional surface of a ball. Lower-dimensional pictures are useful analogies but cannot replace the actual statement.

03 / Progress & scope

Included here as a solved landmark: it shows why a mathematics frontier needs a history of resolved problems as well as a list of open ones.

Try an experiment

Make an abstract question tangible

Keep dimension, compactness and boundary conditions attached to a theorem. Removing one can turn a correct statement into a different problem.

Which evidence supports a general theorem?

Sources & further reading

Clay Mathematics Institute

Sources last checked 2026-10-20. Status reflects an editorial review on that date, not continuous live monitoring.

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